None of the numbers containing an even number in the numerator can be used, as there would be no factor of two in any of the denominators, and therefore any integers produced would be even.

That includes 2/1--it can't be used, so the largest number from the set that can be used is the 5/3. Any number you use is at most 5/3. Even 5/3, when cubed, is less than 5, so you need at least four fractions from the set. But each fraction is more than 3/2, and (3/2)^4 is over five.